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We assume that all realizations are far away from winsorizing with very large \
probability. Nonetheless, since we are solving this numerically, we can \
explicitly account for winsorizing. We do so as follows: there is an \
underlying realization for spending and the signal that is normal with full \
support, but the actual spending is winsorized at the value at which P_3=pmax.\
\>", "Text",
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tbar is the upper bound for taxes; assume that F regime prevails when upper \
bound is insufficient to guarantee p3=pstar, that is for g3>p3star. Also, the \
distribution of g3 is winsorized as discussed above so as to ensure that \
p3<=p3max. \[Sigma] is the variance of the prior. We use tbar and \[Sigma] \
jointly to target the probability that the price level is 2% above target \
(above 1),  and 6% (above target) in period 3; these probabilities are set to \
10% and 5%, respectively\
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Cell["\<\
Computations of the equilibrium in period 2 conditional on no information \
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information in period 2 conditional on no information acquisition in period 1 \
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Posterior mean of the non-winsorized variable, We initialize it on the same \
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P2 inverse under exogenous information (taking winsorizing into account, \
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For future computations, it is convenient to interpolate the second-period \
price (actually, I will interpolate the inverse)\
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Computations of the equilibrium in period 2 conditional on information \
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We use the same vector of posteriors for plots, of course the probability of \
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P2 inverse under information acquisition in period 1 and not in period 2 \
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Computations of the equilibrium in period 1, as well as costs and benefits of \
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The decision not to acquire information when everybody else does is static, \
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P1 inverse under no information acquisition in period 1 (taking winsorizing \
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For future computations, it is convenient to interpolate the price (actually, \
I will interpolate the inverse)\
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compute this, I need the distribution of the posterior mean perceived by the \
informed agents conditional on the information of the uninformed agent. The \
posterior of the uninformed agent, denoted \[Mu]p1 is (\[Tau] \[Mu]+\[Tau]s \
sig1)/(\[Tau]+\[Tau]s). The posterior of the informed agent is (\[Tau] \[Mu]+\
\[Tau]s sig1+\[Tau]scostly \
sig1costly)/(\[Tau]+\[Tau]s+\[Tau]scostly)=((\[Tau]+\[Tau]s)\[Mu]p1+\[Tau]\
scostly sig1costly)/(\[Tau]+\[Tau]s+\[Tau]scostly). From the perspective of \
the uninformed agent, sig1costly=G_3+error of the costly signal (call it \
err1costly). The signal is unbiased, so the error has mean zero (uncorrelated \
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therefore the same as that of G_3, so it is \[Mu]p. For the variance, the \
error is uncorrelated with G_3, so Var(sig1costly|free \
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with the free information, so the precision of that component if \
\[Tau]scostly. The precision of the first piece is \[Tau]p1. The bottom line \
is that the distribution of the expectation held by the informed agents \
conditional on the uninformed information set has a mean \[Mu]p1 and a \
standard deviation corresponding to the standard deviation of the costly \
signal multiplied by the appropriate factor in the expression above. We put \
all of this together in the line below:\
\>", "Text",
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Cell["\<\
We can get an idea of the approximation error introduced by the second \
interpolation step by evaluating the cost without the interpolation in \
costfun1 (this can easily be done at a single point). The approximation is \
excellent\
\>", "Text",
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Cell["\<\
Benefit of acquiring information in period 1 when nobody else does  \
(multiplicative factor). This is made of two pieces: there is the benefit in \
period 1, and the benefit in period 2. We start from the benefit in period 1.\
\>", "Text",
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Cell["\<\
First, we compute the expectation of the period-2 price conditional on the \
expectation that the agent who acquired the costly signal has about the \
expectation that uninformed agents will hold as of period 2. \
\>", "Text",
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Cell["\<\
hatexpp2 is the expectation that the informed agent has about the inverse of \
the second-period price in period 1. This is expressed as a function of the \
information available to the uninformed agent (summarized by their posterior, \
indexed by i) and the information available to the informed agents \
(summarized again by her posterior, indexed by j). The first integral \
computes the expectation conditional on the second-period signal being such \
that the uninformed agents as of period 1 will also choose to remain \
uninformed in period 2. The second integral computes the expectation \
conditional on the second-period signal being such that the uninformed agents \
as of period 1 will pay the cost of acquiring information in period 2, so \
they know the realization of the shock  and p2=p3. For this second part, the \
outer integral is the distribution of G_3 conditional on the information \
known as of period 1 by the informed agent (who paid for the costly signal), \
and the inner integral (the CDF), is the probability that the uninformed \
agents have a posterior mean as of period 2 that exceeds \
criticalbenefitvalue, leading them to acquire the costly signal of period 2. \
This latter probability is conditional on the realization of G_3 (since it is \
inside the outer integral for G_3) and conditional on the signals that the \
informed agent has (in particular, once we condition on G_3, the only \
relevant extra information is the free signal of period 1, represented \
through the posterior mean as of period 1)\
\>", "Text",
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Cell["\<\
As a first step, hat\[Sigma]2 is the standard deviation of the posterior mean \
as of period 2 conditional on free information, computed from the perspective \
of the period-1 informed agent who knows both the free and the costly signal \
of period 1\
\>", "Text",
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We get warnings when integrals are extremely close to 0, which is irrelevant \
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The next piece computes the part of the benefit of unilaterally acquiring \
information in period 1 that accrues in period 2. We start by computing all \
of the appropriate moments that show up in the equation for the profit \
factor, conditional on the information held in period 2 by the agent who \
unilaterally acquired information in period 1 and has received the free \
period-2 signal, but has not yet decided whether to acquire extra information \
in period 2.\
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Next, profit factor when others do not acquire information. This one depends \
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the function in the region in which others acquire information (we keep one \
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represents the period-2 information available to the agent that has deviated \
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1., and the index j represents the information of the agents who know the \
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The next step is to compute moments of the distributions that we need for \
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Cell["\<\
The conditional means are simple, since they are just \[Mu]p1. We can save a \
double integral by taking the outer integral to be hat\[Mu]p2 (conditional on \
\[Mu]p1) and using the formula for a conditional normal distribution to get \
the distribution of \[Mu]p2 conditional on \[Mu]p1 and hat\[Mu]p2. It is \
important to understand this second conditioning: this is not conditional on \
the two free signals and the costly signal of period 1 (because otherwise it \
would be a finer partition and \[Mu]p2 would be known). It is just \
conditioning on knowing the posteriors \[Mu]p1 and hat\[Mu]p2, which are \
appropriate linear combinations of the signals. What we are computing is \
still the double integral conditional on \[Mu]p1, except that ordering the \
integration this way the inner integral is simply a cumulative distribution \
function.\
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Cell["\<\
Get the probability that others will acquire information in period 2, when \
they did not in period 1, conditional on the period-1 freely available \
information and on the posterior mean as perceived by somebody who acquired \
information in period 1 (but not yet in period 2). As usual, when the result \
is extremely close to zero we get warnings, but we only care about the \
probability when it is meaningfully different from zero\
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Cell["Sample paths", "Section",
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Cell["\<\
We now present two sample paths in which initially bad news triggers \
information acquisition. The first sample path represents an \
\[OpenCurlyDoubleQuote]inflation scare:\[CloseCurlyDoubleQuote] news in the \
second period is not as bad, and inflation backtracks. In the second sample \
path, bad news persists into the second period, triggering further \
information acquisition, which is also revealed to be adverse and triggers \
even higher inflation. These paths are chosen purely for illustration.\
\>", "Text",
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We assume that the same mean prevails after the costly signal, which requires \
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In the good sample path, \[Mu]p2goodsamplepath represents the posterior after \
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We then assume that G3 comes exactly at G3=\[Mu]p2goodsamplepath, so period-3 \
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In the bad sample path, we assume that the posterior after the free signal in \
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value below\
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